Comparing Mixed-Integer Programming and Constraint Programming Models for the Hybrid Flow Shop Scheduling Problem with Dedicated Machines

The paper considers a two-stage hybrid flow shop scheduling problem with dedicated machines and release dates. Each job must be first processed on the single machine of stage 1, and then, the job is processed on one of the two dedicated machines of stage 2, depending on its type. Moreover, the jobs are available for processing at their respective release dates. Our goal is to obtain a schedule that minimizes the makespan. This problem is strongly NP-hard. In this paper, two mathematical models are developed for the problem: a mixed-integer programming model and a constraint programming model. The performance of these two models is compared on different problem configurations. And the results show that the constraint programming outperforms the mixed-integer programming in finding optimal solutions for large problem sizes (450 jobs) with very reasonable computing times.

Mixed graph colouring as scheduling multi-processor tasks with equal processing times

A problem of scheduling partially ordered unit-time tasks processed on dedicated machines is formulated as a mixed graph colouring problem, i. e., as an assignment of integers (colours) {1, 2, …, t} to the vertices (tasks) V {ν1, ν2, …, νn}, of the mixed graph G = (V, A, E) such that if vertices vp and vq are joined by an edge [νp, νq] ∈ E their colours have to be different. Further, if two vertices νp and νq are joined by an arc (νi, νj) ∈ A the colour of vertex νi has to be no greater than the colour of vertex νj. We prove that an optimal colouring of a mixed graph G = (V, A, E) is equivalent to the scheduling problem GcMPT|pi = 1|Cmax of finding an optimal schedule for partially ordered multi-processor tasks with unit (equal) processing times. Contrary to classical shop-scheduling problems, several dedicated machines are required to process an individual task in the scheduling problem GcMPT|pi = 1|Cmax. Moreover, along with precedence constraints given on the set V {ν1, ν2, …, νn}, it is required that a subset of tasks must be processed simultaneously. Due to the theorems proved in this article, most analytical results that have been proved for the scheduling problems GcMPT |pi = 1|Cmax so far, have analogous results for optimal colourings of the mixed graphs G = (V, A, E), and vice versa.

Uniform Parallel Machine Scheduling with Dedicated Machines, Job Splitting and Setup Resources

We examine a uniform parallel machine scheduling problem with dedicated machines, job splitting, and limited setup resources for makespan minimization. In this problem, machines have different processing speeds, and each job can only be processed at several designated machines. A job can be split into multiple sections and those sections can be processed on multiple machines simultaneously. Sequence-independent setup times are assumed, and setup operations between jobs require setup operators that are limited. For the problem, we first develop a mathematical optimization model and for large-sized problems a constructive heuristic algorithm is proposed. Finally, we show that the algorithm developed is efficient and provides good solutions by experiments with various scenarios.